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A

lvaro Romero-Calvo

Space Propulsion Laboratory,

Department of Aerospace Science and

Technology,

Politecnico di Milano,

Via Giuseppe La Masa, 34,

Milan 20156, Italy

e-mail: alvaro1.romero@mail.polimi.it

Gabriel Cano Gomez

Departamento de Fı

sica Aplicada III,

Universidad de Sevilla,

Avenida de los Descubrimientos s/n,

Sevilla 41092, Spain

e-mail: gabriel@us.es

Elena Castro-Hernandez

Area de Mecanica de Fluidos,

Dep. Ingenierı

a Aeroespacial y Mecanica de

Fluidos,

Universidad de Sevilla,

Avenida de los Descubrimientos s/n,

Sevilla 41092, Spain

e-mail: elenacastro@us.es

Filippo Maggi

Space Propulsion Laboratory,

Department of Aerospace Science and

Technology,

Politecnico di Milano,

Via Giuseppe La Masa, 34,

Milan 20156, Italy

e-mail: ﬁlippo.maggi@polimi.it

Free and Forced Oscillations of

Magnetic Liquids Under Low-

Gravity Conditions

The sloshing of liquids in microgravity is a relevant problem of applied mechanics with

important implications for spacecraft design. A magnetic settling force may be used to

avoid the highly non-linear dynamics that characterize these systems. However, this

approach is still largely unexplored. This paper presents a quasi-analytical low-gravity

sloshing model for magnetic liquids under the action of external inhomogeneous magnetic

ﬁelds. The problems of free and forced oscillations are solved for axisymmetric geometries

and loads by employing a linearized formulation. The model may be of particular interest

for the development of magnetic sloshing damping devices in space, whose behavior can be

easily predicted and quantiﬁed with standard mechanical analogies.

[DOI: 10.1115/1.4045620]

Keywords: computational mechanics, liquid sloshing, ferrohydrodynamics, microgravity

1 Introduction

The term sloshing refers to the forced movement of liquids in par-

tially ﬁlled tanks [1]. Propellant sloshing has been a major concern

for space engineers since the beginning of the space era. During

launch, it can result in the partial or total loss of control of the space-

craft [2]. In a low-gravity environment, the liquid tends to adopt a

random position inside the tank and mixes with pressurizing gas

bubbles. This results in a complicated propellant management

system design, often increasing the inert mass of the vehicle [1].

Low-gravity sloshing is characterized by the dominant role of

surface tension that produces a curved equilibrium free surface

(or meniscus) and a complex interaction with the walls of the

vessel that contains the liquid. The ﬁrst solution of the low-gravity

free surface oscillation problem was given in 1964 by Satterlee and

Reynolds for cylindrical containers [3]. In the context of the Space

Race, a signiﬁcant effort was made to study low-gravity sloshing in

cylindrical [3–9], spheroidal [8,10,11], or axisymmetric [12–15]

tanks. A non-extensive list of modern works includes numerical

models for cryogens [16,17], coupled non-linear implementations

[18], or computational ﬂuid dynamics (CFD) simulations [19]. Ana-

lytical solutions of the free and forced oscillations problem were

found by Utsumi [20–23].

Different active and passive strategies have been traditionally

employed to mitigate liquid sloshing in microgravity. Active

approaches settle the propellant by imposing an adequate inertial

force with a set of thrusters. Passive techniques make use of

surface tension or membranes to hold the liquid at a certain position

and reduce the effect of random accelerations. The resulting techni-

cal implementations, named Propellant Management Devices, are

currently used to grant adequate liquid propellant feeding in case

of in-orbit ignition of chemical propulsion units [24].

Since the absence of a settling volume force is the main character-

istic of low-gravity sloshing, the problem could be attacked by repro-

ducing the force of gravity with electromagnetic ﬁelds if the liquid

can answer to such stimulus. The use of dielectrophoresis, a phenom-

enon on which a force is exerted on dielectric particles in the presence

of a non-uniform electric ﬁeld, was explored by the US Air Force

with dielectric propellants in 1963 [25]. The study unveiled a high

risk of arcing inside the tanks and highlighted the need for large,

heavy, and noisy power sources. Approaches exploring Magnetic

Positive Positioning have also been suggested to exploit the inherent

magnetic properties of paramagnetic (oxygen) and diamagnetic

(hydrogen) liquids. Relatively recent studies employed numerical

simulations and microgravity experiments to validate this concept

[26,27].

Ferroﬂuids are colloidal suspensions of magnetic nanoparticles

treated with a surfactant to prevent from agglomeration. As a

result, they exhibit high magnetic susceptibility. Their invention is

attributed to Steve Papell, who in 1963 proposed to “provide an arti-

ﬁcially imposed gravity environment”with ferroﬂuid-based mag-

netic propellants [28]. The basic equations governing the dynamics

of ferroﬂuids were presented in 1964 by Neuringer and Rosensweig

[29], giving rise to the ﬁeld of Ferrohydrodynamics [30]. Although

since then ferroﬂuids have found numerous applications on Earth,

works addressing their original purpose are scarce. A rare exception

is the NASA Magnetically Actuated Propellant Orientation experi-

ment, which studied the magnetic positioning of liquid oxygen and

validated a custom CFD model with a series of parabolic ﬂight

Contributed by the Applied Mechanics Division of ASME for publication in the

JOURNAL OF APPLIED MECHANICS. Manuscript received August 26, 2019; ﬁnal manu-

script received December 3, 2019; published online December 6, 2019. Assoc.

Editor: N. R. Aluru.

Journal of Applied Mechanics FEBRUARY 2020, Vol. 87 / 021010-1Copyright © 2019 by ASME

experiments with ferroﬂuids [31]. Subsequent publications pre-

sented numerical models to study and generalize the measurements

for space applications [32–34].

One of the main drawbacks of magnetic sloshing control is the

rapid decay of magnetic ﬁelds, which limits its applicability to

small and compact tanks. In this context, the increasing number of

propelled microsatellites may beneﬁt from this technology since a

direct control of liquid sloshing could be achieved with small

low-cost magnets. Once the liquid is positioned, magnetic ﬁelds

could also be used to tune the natural frequencies and damping

ratios of the system. This approach has been adopted for terrestrial

applications, such as tuned magnetic liquid dampers [35,36].

On-ground research exploring axisymmetric sloshing [37,38], the

frequency shifts due to the magnetic interaction [39], two-layer

sloshing [40], or the swirling phenomenon [41], among others, has

been carried out in the past with notable results. The sloshing of fer-

roﬂuids in low-gravity was indirectly studied in 1972 with a focus on

gravity compensation techniques [42].

This work addresses the free and forced oscillations of magnetic

liquids in axisymmetric containers when subjected to an external

inhomogeneous magnetic ﬁeld in microgravity. A ferrohydrody-

namic model is developed to predict the natural frequencies and

modal shapes of the system and a case of application is presented.

Unlike non-magnetic low-gravity sloshing, the presence of a restor-

ing force ensures that the hypothesis of small oscillations (linear

sloshing) is satisﬁed in a wide range of operations.

2 Problem Formulation

The system to be modeled is represented in Fig. 1. A volume Vof a

magnetic liquid ﬁlls an upright axisymmetric tank with radius aat the

meniscus contour. The liquid is incompressible, Newtonian and is

characterized by a density ρ, speciﬁc volume v=ρ

−1

, kinematic vis-

cosity ν, surface tension σ, and magnetization curve M(H). Hand M

are, respectively, the modules of the magnetic ﬁeld Hand magneti-

zation ﬁeld M, which are assumed to be collinear. The liquid

meets the container wall with a contact angle θ

c

. An applied inhomo-

geneous axisymmetric magnetic ﬁeld H

0

is imposed by an external

source (e.g., a coil located at the base of the container). The inertial

acceleration galong the z-axis is also considered. A non-reactive

gas at pressure p

g

ﬁlls the free space. In the ﬁgure, sis a curvilinear

coordinate along the meniscus with origin in the vertex Oand the rel-

ative heights are given by w(ﬂuid surface–vertex), f(meniscus–

vertex) and h(ﬂuid surface–meniscus). The container is subjected

to a lateral displacement x(t). The meniscus is represented by a

dashed line, and the dynamic ﬂuid surface is given by a solid line.

The model here presented extends the works by Satterlee and

Reynolds [3] and Yeh [12] by considering the magnetic interaction

and the axisymmetric oscillations case.

2.1 Nonlinear Formulation. A cylindrical reference system

{u

r

,u

θ

,u

z

}, centered at the vertex of the meniscus, is subsequently

considered. If an irrotational ﬂow ﬁeld is assumed, there exists a

potential φsuch that

v=−∇φ=−φrur−1

rφθuθ−φzuz(1)

being vthe ﬂow velocity with the subindices denoting the deriva-

tive. The velocity potential satisﬁes Laplace’s equation

∇2φ=φrr +φr

r+φθθ

r2+φzz =0inV(2)

subjected to the non-penetration wall boundary condition

φr=−˙

xcos θ,φθ/r=˙

xsin θ,φz=0onW(3)

An additional boundary condition at the free surface is given by the

unsteady ferrohydrodynamic Bernoulli’s equation, which for an iso-

thermal system with collinear magnetization ﬁeld Madopts the

form [30,43]

−˙

φ+v2

2+p*

ρ+gw −ψ

ρ+˙

xcos θφr−˙

xsin θφθ

r

=β(t)onS(4)

where gis the inertial acceleration, ψis the magnetic force potential,

β(t) is an arbitrary function of time, and p* is the composite pres-

sure,deﬁned as [30]

p*=p(ρ,T)+μ0H

0

v∂M

∂v

H,T

dH+μ0H

0

M(H)dH(5)

with the ﬁrst, second, and third terms being named thermodynamic,

magnetostrictive, and ﬂuid-magnetic pressure, respectively. For

magnetically diluted systems M∼ρ, where ρis the concentration

of magnetic particles for the case of ferroﬂuids. Under this addi-

tional assumption, both pressure-like components are approxi-

mately compensated, and hence p*≈p(ρ,T)[30].

The canonical magnetic force per unit volume is given by

μ0M∇H, with μ

0

=4π·10

−7

N/A

2

being the permeability of free

space [30]. It can be shown that, for an isothermal ﬂuid, this

force derives from the potential [29]

ψ=μ0H

0

M(H)dH(6)

Due to the discontinuity of the Maxwell stress tensor at the mag-

netic liquid interface, the ferrohydrodynamic boundary condition

in the absence of viscous forces becomes

p*=pg−pc−pnon S(7)

where pn=μ0M2

n/2 is the magnetic normal traction,M

n

is the mag-

netization component normal to the ﬂuid surface, and p

c

is the cap-

illary pressure. The last is deﬁned by the Laplace-Young equation

p

c

=σK, where

K=1

r

∂

∂r

rwr

1+w2

r+1

r2w2

θ

⎡

⎢

⎢

⎣⎤

⎥

⎥

⎦

+1

r2

∂

∂θ

wθ

1+w2

r+1

r2w2

θ

⎡

⎢

⎢

⎣⎤

⎥

⎥

⎦

(8)

is the curvature of the surface [1]. Since at Eq. (4) only the spatial

derivatives of the velocity potential have a physical meaning (e.g.,

Fig. 1 Geometry of the system under analysis, composed of a

magnetic liquid that ﬁlls a container in microgravity while sub-

jected to an external magnetic ﬁeld. S′and C′refer to the menis-

cus surface and contour, while S and C are the dynamic ﬂuid

surface and contour, respectively. O denotes the vertex of the

meniscus, W is the vessel wall, and V denotes the ﬂuid volume.

021010-2 / Vol. 87, FEBRUARY 2020 Transactions of the ASME

Eq. (1)), any function of time can be added to φif mathematically

convenient. From a physical viewpoint, the absolute value of p

remains undetermined under the incompressible ﬂow assumption

[43]. The integration constant β(t) can be then absorbed into the def-

inition of φ. By arbitrarily selecting β(t)=p

g

/ρ, the dynamic inter-

face condition is obtained

˙

φ−1

2φ2

r+1

r2φ2

θ+φ2

z

+σ

ρK−gw +ψ

ρ+μ0M2

n

2ρ

−˙

xcos θφr+˙

xsin θφθ

r=0onS

(9)

In an inertial reference system, the vertical displacement wof a

surface point lying at (r,θ) in the interface z=w(r,θ,t) is given by

dw

dt=˙

w+wr

dr

dt+wθ

dθ

dton S(10)

If the velocity components relative to the tank dw/dt,dr/dt, and

rdθ/dtare expressed as a function of the potential given by

Eq. (1), the kinematic interface condition that relates the last with

the shape of the free surface is

˙w=−φz+wrφr+˙xcos θ

+wθ

r2φθ−˙xr sin θ

on S(11)

The continuity equation given by Eq. (2), the kinematic relation in

Eq. (11) and the boundary conditions in Eqs. (3) and (9),deﬁne the

problem to be solved after imposing the contact angle at the wall

(θ

c

) and a contact hysteresis parameter that will be described later

in the text.

2.2 Equilibrium Free Surface Shape. Due to the axisymme-

try of geometry and loads, the static equilibrium surface of the ﬂuid

(S′) is also axisymmetric. Its shape can be determined from the

balance of vertical forces in a circular crown of inner radius rand

inﬁnitesimal width dr along the surface [24]. This results in the fol-

lowing set of dimensionless differential equations:

d

dSRdF

dS

=RdR

dSλ+BoF −ψ(R)

(12a)

dF

dS

d2F

dS2+dR

dS

d2R

dS2=0(12b)

and boundary conditions

R(0) =F(0) =dF(0)

dS=0,dR(0)

dS=1(12c)

dF(1)

dR=tan π

2−θc

(12d)

where R=r/a,F=f/a,S=s/a,Bo =ρga

2

/σis the Bond number,

λ=a(p

g

−p

0

)/σ, being p

0

the liquid pressure at the free surface

vertex, ψincludes the magnetic potential and magnetic normal trac-

tion through

ψ(R)=aμ0

σH(R,F(R))

H(0,0)

M(H)dH+M2

n

2

F(R)

(13)

and the static contact angle with respect to the vertical θcis given by

θc=θc+π

2−arctan dW

drC′

(14)

A numerical solution can be easily computed by (1) setting an initial

vertex position, (2) calculating the value of λiteratively in order to

satisfy the contact angle condition given by Eq. (12d), (3) solving

the system with an ODE solver, and (4) obtaining the new height

of the vertex through volume conservation. The procedure is

repeated until the vertex height converges with a prescribed relative

variation. When non-trivial magnetic setups are involved, a FEM

simulation must be included in the loop.

2.3 Linear Equations. The dynamic and kinematic conditions

in Eqs. (9) and (11) are highly nonlinear. The standard analytical

procedure overcomes this difﬁculty by linearizing the problem

and restricting the analysis to small oscillations. If the wave position

is expressed as the sum of the static equilibrium solution and a small

perturbation

w(r,θ,t)=f(r)+h(r,θ,t)(15)

it will be possible to express the system of equations and boundary

conditions as a Taylor’s series expansion around the equilibrium

surface S′. If second-order terms are neglected, the boundary-value

problem becomes

∇2φ=0inV(16a)

φr=−˙

xcos θ,φθ/r=˙

xsin θ,φz=0onW

˙

φ+σ

ρ

1

r

∂

∂r

rhr

1+f2

r

3/2

+1

r2

∂

∂θ

hθ

1+f2

r

−g−μ0

ρM∂H

∂z+Mn

∂Mn

∂z

h=0onS′

(16b)

˙

h=−φz+fr(φr+˙

xcos θ)onS′(16c)

hr=γhon C′(16d)

Equation (16d)assumes that the slope of the perturbation ﬁeld at

the wall is related to the magnitude of the perturbation at the same

point through the parameter γ. The free-edge condition is character-

ized by γ=0, while the stuck-edge condition is characterized by

γ→∞[12]. This assumption is far from being rock-solid and has

indeed motivated a strong debate in the past. It has been suggested

that the contact angle hysteresis condition depends not only on the

position of the wave but also on its velocity [4] or the state of the

wall [7]. In the absence of a clear criteria, some studies assume

the free-edge condition or intermediate approaches, generally

obtaining a reasonable agreement with experimental data [10].

The only difference between the previous formulation and the

classical problem without magnetic interactions is given by the

magnetic term in Eq. (16b). The effective gravity acceleration

includes both inertial and magnetic components and is given by

g*(r)=g−μ0

ρM∂H

∂z+Mn

∂Mn

∂z

S

(17)

where it can be observed that the magnetic contribution at the

surface is a function of the radius. The magnitude and relative

importance of the magnetic terms depend on the magnetic conﬁgu-

ration and gravity level of the system under analysis. In particular,

the magnetic component will be more signiﬁcant in the absence of

gravity.

The magnetic ﬁeld modiﬁes the effective gravity acceleration of

the system and shifts its natural frequencies, as reported in normal-

gravity works [35,44]. If the magnetic term was approximately

constant in R, like in the case of a linear magnetic ﬁeld and a ﬂat

surface, the problem would be equivalent to the non-magnetic

system [42]. In this analysis, however, an inhomogeneous magnetic

ﬁeld is considered.

Journal of Applied Mechanics FEBRUARY 2020, Vol. 87 / 021010-3

2.4 Extraction of Tank Motion. The potential φhas been

referred to an inertial reference system. In order to analyze the

movement of the free surface in the tank reference frame, it

would be convenient to split this potential into rigid body (ϕ

0

)

and perturbed (ϕ) components, so that

φ=ϕ0+ϕ,ϕ0=−˙

xrcos θ(18)

The boundary-value problem can be then expressed as a function of

the perturbed potential and becomes

∇2ϕ=0inV(19a)

ϕn=0onW(19b)

˙

ϕ+σ

ρ

1

r

∂

∂r

rhr

1+f2

r

3/2

+1

r2

∂

∂θ

hθ

1+f2

r

−g−μ0

ρM∂H

∂z+Mn

∂Mn

∂z

h=¨

xrcos θon S′

(19c)

˙

h=−ϕz+ϕrfron S′(19d)

hr=γhon C′(19e)

3 Free Oscillations Problem

3.1 Dimensionless Linear Equations. In Refs. [3,12], it is

proposed to split the potentials ϕand hinto spatial and temporary

components, the second being a cyclic function of time with a cir-

cular frequency ω. The resulting dimensionless boundary-value

problem is

∇2Φ=0 inV (20a)

∂Φ

∂n=0onW(20b)

Ω2Φ−[Bo +Bomag(R)]H

+1

R

∂

∂R

RHR

1+F2

R

3/2

+1

R2

∂

∂θ

Hθ

1+F2

R

=0onS

′

(20c)

H=ΦZ−ΦRFRon S′(20d)

HR=ΓHon C′(20e)

where R=r/a,Z=z/a,F=f/a,ϕ(R,θ,Z,t)=

g0a3

Φ(R,θ,Z)

sin(ωt), h(R,θ,t)=

ag0/ω2

H(R,θ) cos(ωt), Ω

2

=ρa

3

ω

2

/σ,Γ=

aγ, and g

0

is the acceleration of gravity at ground level [1]. The

Magnetic Bond Number has been deﬁned as

Bomag(R)=−μ0a2

σM∂H

∂z+Mn

∂Mn

∂z

F(R)

(21)

and accounts for the effects of the external magnetic ﬁeld on the

liquid.

3.2 Variational Formulation. By following the procedure

described in Refs. [3,12], it is possible to develop a variational prin-

ciple equivalent to Eqs. (20b)and (20c)as

I=S′

H2

R

1+F2

R

3/2+1

R2

H2

θ

1+F2

R

1/2

+Bo +Bomag(R)

H2−Ω2ΦHRdRdθ

−Ω2W

ΦGR dRdθ−ΓC′

H2

1+F2

R

3/2

R=1

dθ

=extremum

(22a)

subjected to

∇2Φ=0inV(22b)

H=ΦZ−FRΦRon S′(22c)

G=ΦZ−WRΦRon W (22d)

HR=ΓHon C′(22e)

where Gand its associated terms arise from the application of the

wall boundary condition given by Eq. (20b)as detailed in

Ref. [12]. The obtention of this variational formulation follows

the procedure described in Refs. [12,45].

3.3 Ritz Method. The previous set of equations can only be

analytically solved for simpliﬁed conﬁgurations in the absence of

magnetic ﬁelds, like the case of a cylindrical container with a ﬂat

bottom and ﬂat ﬂuid surface (θc=90 deg) [3]. For other physical

systems, Ritz approximations [1,12]orﬁnite differences approaches

[5,6] have been proposed to compute the eigenfunctions of the

problem. The basic formulation of the ﬁrst approach is subsequently

developed based on Refs. [3,12].

By following Ritz’s method, the eigenfunctions Φ

(n)

can be

approximated as the linear combination of admissible functions

Φi(R,θ,Z) that satisfy the boundary conditions of the problem

described by Eqs. (22b)–(22e). This results in

Φ(n)=

N

i=1

C(n)

iΦi(n=1,...,N)(23)

where Nis the size of the set of admissible functions. In the same

way, the eigenfunctions H(n)

iand G

(n)

are approximated by

ζi(R,θ) and ξi(R,θ) through

H(n)=

N

i=1

C(n)

iζi(24)

G(n)=

N

i=1

C(n)

iξi(25)

The sets of admissible functions are linked through Eqs. (22c)and

(22d).IfΦ

(n)

,H(n), and G

(n)

are continuous functions of C(n)

i, the

extremum condition represented by Eq. (22a)requires that

∂I

∂C(n)

i

=0,(i=1,2,...,N)(26)

which results in the system of equations

N

i=1

C(n)

iRij +BoLij +Lmag

ij −Ω2

nQij

=0,

(j=1,2,...,N)

(27)

021010-4 / Vol. 87, FEBRUARY 2020 Transactions of the ASME

being

Rij =F

ζiRζjR

(1 +F2

R)3/2+n2ζiζj

R2(1 +F2

R)1/2

RdRdθ

−Γ2π

0

ζiζj

(1 +F2

R)3/2

R=1

dθ

(28a)

Lij =F

ζiζjRdRdθ(28b)

Lmag

ij =F

Bomag(R)ζiζjRdRdθ(28c)

Qij =1

2F

Φiζj+Φjζi

RdRdθ

+1

2F

Φiξj+Φjξi

RdRdθ

(28d)

The system has a nontrivial solution only when its determinant is

zero. The eigenvalues Ω2

n, and therefore the corresponding modal

circular frequencies ω

n

, are then computed by means of the charac-

teristic equation:

Rij +BoLij +Lmag

ij −Ω2Qij=0(29)

Once solved, the eigenfunctions of the problem are obtained from

Eq. (23) to Eq. (25).

3.4 Forced Lateral Oscillations and Mechanical Analogies.

In order to solve the forced lateral oscillations case, a modal solu-

tion for the linearized boundary-value problem given by Eq. (19)

is built from the eigenmodes Φ

(n)

and H(n). This solution is

expressed as

ϕ=

N

n=1

An(t)Φ(n),h=

N

n=1

Bn(t)H(n)(30)

where the modal coordinates A

n

(t) and B

n

(t) are computed from

their corresponding modal equations. The reader is referred to

Ref. [12] for a full description of this procedure.

Since the modal equations are linear, the forced sloshing problem

can be conceived as the superposition of several linear oscillators.

These are usually assumed to be a series of spring-mass systems

on which a linear damper is included a posteriori. The employment

of a mechanical analogy simpliﬁes the integration of the sloshing

problem into the equations of motion of the vehicle. In the case

here analyzed, an additional magnetic pressure should be consid-

ered. However, in virtue of Newton’s action-reaction principle, if

the magnetic source is rigidly coupled to the tank then the distribu-

tion of magnetic pressures cannot produce torque in the assembly.

That is, non-magnetic mechanical analogies can be extended for

the magnetic case by simply employing the new magnetic eigen-

modes Φ

(n)

and H(n)with their corresponding eigenfrequencies.

Some possibilities are the models developed by Dodge and Garza

[46] or Utsumi [23].

3.5 Selection of Admissible and Primitive Functions. The

set of admissible functions for Φ,H, and G, related through

Eqs. (22c)and (22d)satisfy by deﬁnition Eqs. (22b)–(22e)and

form truncated series that approximate the eigenfunctions of the

problem. A set of primitives should be previously deﬁned as [3]

ϑp=Jn(kpR) cos(mθ)ekpZ(p=1,...,N,N+1) (31a)

ζp=ϑpZ −FRϑpR

Z=F(R)(31b)

ξp=ϑpZ −WRϑpR

Z=W(R)(31c)

with k

p

being the roots of the equation

d

dRJn(kiR)

R=1

=0(32)

where J

n

is the Bessel function of ﬁrst kind and order n. This index

is used to study the axisymmetric (n=0) and lateral (n=1) cases,

while mdeﬁnes the circumferential symmetry of the problem. Axi-

symmetric primitive functions will be characterized by n=m=0,

while lateral sloshing functions will be characterized by n=m=1.

However, the previous set of primitives does not satisfy

Eq. (22e). The set of admissible functions is then created as a

linear combination of the previous

Φi=

N+1

p=i

aipϑp,ζi=

N+1

p=i

aipζp,ξi=

N+1

p=i

aipξp,(i=1,2,...,N)

(33)

The N+1−pcoefﬁcients a

ip

for each ivalue are determined by

imposing (i) a normalization condition, (ii) a contact angle value,

and (iii) a Lagrange minimization problem designed to produce

Bessel-like functions. These conditions are, respectively, expressed

as [3]

N+1

p=i

aipζp(1) =1(34a)

N+1

p=i

aipζpR (1) =Γ(34b)

N+1

p=i

aip(Kpj −k2

pLpj)+λ1iζjR (1)

+λ2iζj(1) =0(j=i,i+1,...,N,N+1)

(34c)

where λ

1i

and λ

2i

are the Lagrange multipliers of the minimization

problem and

ζi(R)=ζi(R,θ)/cos(mθ)(35)

Kij =F

ζiRζjR −n

R2ζiζj

RdRdθ(36)

Lij =F

ζiζjRdRdθ(37)

Once the system is solved, the admissible set can be used to solve

the eigenvalue problem.

The success of this method depends on ﬁnding an adequate set of

admissible functions Φisuch that the eigenfunctions Φ

(n)

can be

represented with a reduced number of elements. The Zterm in the

primitives ϑ

p

, evaluated at the equilibrium surface, grows exponen-

tially when F(R) departs signiﬁcantly from Z=0. This is the case of

low Bond numbers and small contact angles. In Ref. [13], it is stated

that for contact angles lower than 15 deg in the case of free-edge

condition (Γ=0) or lower than 60 deg for the stuck-edge condition

(Γ→∞), the system may become numerically ill-conditioned. Fur-

thermore, the comparison between this method and a ﬁnite differ-

ences approach showed signiﬁcant divergences in the shape of

the eigenfunctions Φ

(n)

for particular cases.

A potential solution would be ﬁnding a set of primitive functions

without exponential terms. To the best knowledge of the authors

Journal of Applied Mechanics FEBRUARY 2020, Vol. 87 / 021010-5

and considering the attempts made in Ref. [3], an alternative has not

yet been proposed. It should also be noted that the magnetic force

generally ﬂattens the equilibrium surface, hence mitigating the

effect of the exponential term in Eq. (31a).

4 Case of Application

4.1 System Description. CubeSats are a particular class of

nanosatellites composed of standardized 10 × 10 cm cubic units

(U) [47]. Their subsystems are designed to ﬁt this standard, lower-

ing the manufacturing costs and enhancing adaptability (e.g., the

Aerojet MPS-130 propulsion module is offered in 1U or 2U

formats). Due to the rapid decay of magnetic ﬁelds with distance,

small spacecrafts with small propellant tanks may be particularly

well suited for magnetic sloshing control implementations. Their

development requires a dedicated feasibility study that is beyond

the scope of this paper; however, the performance of a hypothetical

system is here addressed.

A 1U cylindrical container with 10 cm height, 5 cm radius, and

ﬁlled with a ferroﬂuid solution up to half of its height is subse-

quently considered. In order to generate a downward restoring

force, a high-end cylindrical neodymium magnet magnetized at

1500 kA/m in the vertical direction is placed under the tank. The

magnet has a hole of r

i

=2.5 mm radius at the center (liquid

outlet), a height h

m

, an external radius r

e

, and a density ρ

m

=

7010 kg/m

3

. A contact angle of 67 deg is assumed in microgravity

conditions. The sketch of this setup is given in Fig. 2.

The liquid is a 1:10 volume solution of the Ferrotec water-based

EMG-700 ferroﬂuid with a 0.58% volume concentration of mag-

netic nanoparticles. Its magnetic properties were measured with a

MicroSense EZ-9 Vibrating Sample Magnetometer at the Physics

Department of Politecnico di Milano. The corresponding magneti-

zation curve is depicted in Fig. 3and shows an initial susceptibility

χ=0.181 and saturation magnetization M

s

=3160 A/m. Viscosity

is assumed to have a negligible effect on the free sloshing problem.

4.2 Magnetic Modeling. The magnetic system is modeled in

COMSOL MULTIPHYSICS, which is interfaced with the model developed

in Sec. 2. The FEM simulation is employed to estimate the ﬁelds H

and Mfor a given magnet and equilibrium surface shape (menis-

cus). The last is computed iteratively by means of Eq. (12) with a

FEM-in-the-loop implementation. Equation (21) is then employed

to calculate the magnetic Bond number at the surface,

which determines the solution of the free oscillation problem. The

eigenfrequencies and eigenmodes of the system are ﬁnally obtained

by solving Eq. (22).

To simulate the magnetic ﬁeld, the model solves the stationary

non-electric Maxwell equations

∇×H=0(38)

B=∇×A(39)

where Ais the magnetic vector potential produced by the magne-

tized materials. The constitutive relation

B=μ0H+M() (40)

is applied to the magnet with M=[0, 0, 1500] kA/m and to the sur-

rounding air with M=0. The magnetization curve M=f(H)in

Fig. 3is applied to the ferroﬂuid volume.

The simulation domain is a rectangular 1 × 2 m region enclosing

the container. An axisymmetric boundary condition is applied to the

symmetry axis, while the tangential magnetic potential is imposed

at the external faces through n×A=n×A

0

.A

0

is the dipole term

of the magnetic vector potential generated by the magnetization

ﬁelds of the magnet and ferroﬂuid. Consequently, A

0

is computed

as the potential vector generated by two point dipoles applied at

the centers of the magnetization distributions and whose dipole

moments are those of said distributions. While the dipole associated

with the magnet can be calculated beforehand, the ferroﬂuid dipole

needs to be approximated iteratively by integrating Min the ferro-

ﬂuid volume. The relative error in the magnetic vector potential due

the dipole approximation is estimated to be below 0.03% at the

boundary of the domain with respect to the exact value generated

by the equivalent circular loop.

The mesh is composed of 53,000 irregular triangular elements

and is reﬁned at the meniscus, as shown in Fig. 4. Mean and

minimum condition numbers of 0.983 and 0.766 are measured.

Figure 5shows a particular conﬁguration of analysis. Positive

and negative curvatures are observed at the meniscus due to the

high intensity of the magnetic ﬁeld. Weaker magnets would result

in convex equilibrium surfaces, as in the non-magnetic case. It

should be noted that the magnetic Bond number rapidly decreases

with distance to the source, spanning between 0 and 20 at the free

surface.

4.3 Parametric Analysis. Figure 6depicts the fundamental

sloshing frequency ω

1

, corresponding to the lowest root of

Eq. (29), as a function of the external radius r

e

and height h

m

of

the magnet. N=7 admissible functions were employed in the com-

putation. The mass of the magnet is given in a second scale, reﬂect-

ing the technical trade-off between mass and sloshing frequency.

Larger magnets result in stronger restoring forces and higher

Fig. 2 Sketch of the case of application. Units in millimeters.

Fig. 3 Magnetization curve of the 1:10 solution of the Ferrotec

EMG-700 water-based ferroﬂuid

021010-6 / Vol. 87, FEBRUARY 2020 Transactions of the ASME

sloshing frequencies. For example, a downward 7 N force and a

100% increase in the fundamental sloshing frequency can be

achieved with a 3 mm height, 30 mm external radius, and 60 g

magnet. Unlike the non-magnetic case, the presence of a signiﬁcant

restoring force ensures that the assumption of small oscillations

(linear sloshing) is not violated for moderate displacements of the

container.

A small ﬂuctuation in the frequency plot appears due to numeri-

cal errors. The procedures for solving Eq. (12) are highly dependent

on the initial estimation of λ. This behavior has been extensively

reported in the bibliography [3,24] and is further complicated by

the magnetic interaction. In addition, the eigenvalue problem that

provides the natural frequencies of the system becomes ill-

conditioned if the actual modal shapes diverge signiﬁcantly from

the primitive functions [13]. For strong magnetic ﬁelds, this may

certainly be the case. The problem would be solved if a ﬁnite differ-

ences approach is employed instead of Ritz’s method.

The modal shapes for the 3 mm height and 30mm external radius

magnet are represented in Fig. 7together with the non-magnetic

modes (i.e., the ones obtained when the magnet is removed).

Although the proﬁles are essentially the same, it is interesting to

observe how the fundamental mode slightly reduces and increases

its vertical displacement where the magnetic Bond number is

greater and smaller, respectively. This is consistent with the afore-

mentioned stabilizing role of the magnetic force.

5 Conclusions

A quasi-analytical model has been developed to study the slosh-

ing of magnetic liquids in low-gravity conditions. The magnetic

interaction modiﬁes the shape of the meniscus and the effective

inertial acceleration of the system as shown in Eqs. (13) and (17),

Fig. 4 Mesh and overall dimensions of the magnetic FEM simu-

lation domain. Units in millimeters.

Fig. 5 Equilibrium conﬁguration of the magnetic sloshing

damping system for h

m

=5 mm and r

e

=2.8 cm. The magnetic

Bond number is represented in the color scale for the range of

interest.

Fig. 6 Fundamental sloshing frequency ω

1

(top) and mass of the

magnet (bottom) as a function of the height h

m

and external

radius r

e

of the magnet

Fig. 7 Magnetic and non-magnetic sloshing modes shape for

the case under analysis

Journal of Applied Mechanics FEBRUARY 2020, Vol. 87 / 021010-7

respectively. As a consequence, a shift of the eigenfrequencies and a

modiﬁcation of the eigenmodes of the free oscillations problem is

produced. The framework here presented extends the models devel-

oped in previous works [3,12] by adding the magnetic and axisym-

metric cases.

The small oscillations assumption is generally not valid for non-

magnetic sloshing in microgravity, which is usually characterized

by complex non-linear deformations driven by surface tension. In

the magnetic case, however, the linear treatment of the problem is

endorsed by a signiﬁcant magnetic restoring force. It would be rea-

sonable to ask how strong magnetic ﬁelds affect the shape of the

eigenmodes and hence the reliability of Ritz’s approximation.

Since this procedure has been historically discussed [13], the devel-

opment of a ﬁnite differences model becomes particularly conve-

nient for future implementations. Ritz’s method represents,

however, the simplest tool to solve the variational formulation

given by Eq. (22) and has been presented for illustrative purposes.

From a technical perspective, the magnetic sloshing concept rep-

resents an opportunity to develop new sloshing control devices for

microsatellites. Unlike the non-magnetic case, the response of the

system can be easily predicted, quantiﬁed, and simulated by

means of standard mechanical analogies. These simpliﬁed models

can be easily embedded in a controller (e.g., a linear observer)

used to predict and compensate the sloshing disturbances of a

spacecraft in orbit. The spacecraft would then beneﬁt from a signif-

icantly improved pointing performance.

Acknowledgment

The authors thank their institutions, Politecnico di Milano and the

University of Seville, for their ﬁnancial and academic support. The

discussions with Prof. Miguel Ángel Herrada Gutiérrez on the ver-

iﬁcation of the non-magnetic model were highly appreciated.

Nomenclature

a=axisymmetric tank radius at the contour of the

meniscus

f=relative height between meniscus and vertex

g=inertial acceleration

h=relative height between meniscus and dynamic

liquid surface

p=thermodynamic pressure

s=curvilinear coordinate along the meniscus

v=speciﬁc volume

w=relative height between dynamic liquid surface and

vertex

x=lateral displacement of the container

n=unitary external vector normal to the ﬂuid surface

v=liquid velocity

C=dynamic contour

F=dimensionless f

G=wall boundary condition function

I=variational principle

K=curvature of the liquid surface

N=size of the set of admissible functions

O=vertex of the meniscus

S=dynamic ﬂuid surface

V=liquid volume

W=walls of the container

Z=dimensionless z

A=magnetic vector potential

B=magnetic ﬂux density

H=magnetic ﬁeld

M=magnetization ﬁeld

H=dimensionless h

S=dimensionless s

a

ip

=modal coefﬁcients used by Φi,ζi, and ξi

g

0

=gravity acceleration at ground level

h

m

=height of the cylindrical magnet

p

0

=thermodynamic pressure at the vertex of the

meniscus

p

c

=capillary pressure

p

g

=ﬁlling gas pressure

p

n

=magnetic normal traction

r

e

=external radius of the cylindrical magnet

r

i

=internal radius of the cylindrical magnet

A

n

=forced problem coefﬁcients for ϕ

B

n

=forced problem coefﬁcients for h

M

n

=magnetization component normal to the ﬂuid

surface

Q

ij

=matrix from Eq. (29)

R

ij

=matrix from Eq. (29)

p*=composite pressure

C′=Meniscus contour

S′=Meniscus surface

G

(n)

=eigenfunctions of G

C(n)

i=modal coefﬁcients used by Φ

(n)

,H(n), and G

(n)

Bo =bond number

Bo

mag

=magnetic bond number

w=relative height between dynamic liquid surface and

vertex for a particular surface point

A

0

=dipole term of A

H

0

=applied magnetic ﬁeld

Lij =matrix from Eq. (29)

Lmag

ij =matrix from Eq. (29)

H(n)=eigenfunctions of H

β=arbitrary time constant of Bernoulli’s equation

Γ=dimensionless γ

γ=surface hysteresis parameter

ζ

p

=primitive functions of ζi

ζi=admissible functions of H(n)

ϑ

p

=primitive functions of Φi

θ

c

=surface contact angle

θc=surface contact angle referred to the vertical

λ=equilibrium free surface parameter

μ

0

=magnetic permeability of free space

ν=kinematic viscosity

ξ

p

=primitive functions of ξi

ξi=admissible functions of G

(n)

ρ=liquid density

σ=surface tension

φ=liquid velocity potential

ϕ

0

=rigid-body liquid velocity potential

ϕ=perturbed liquid velocity potential

Φ=dimensionless ϕ

Φ

(n)

=eigenfunctions of Φ

Φi=admissible functions of Φ

(n)

χ=magnetic susceptibility

ψ=magnetic force potential

ψ=dimensionless magnetic term at the meniscus

Ω=dimensionless ω

Ω

n

=dimensionless ω

n

ω=circular frequency of the surface wave

ω

n

=modal circular frequency of the surface wave

{r,θ,z}=cylindrical coordinates of the system {u

r

,u

θ

,u

z

}

{u

r

,u

θ

,u

z

}=cylindrical reference system centered at the vertex

of the meniscus

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